Here is a fun thought experiment: How highly efficient do you want your fire department?
Henrik Mårtensson has a nice discussion of how an extreme focus on "cost effectiveness" can severely damage an organization with this mindset. He includes a great logic tree of how this works together (thumbnail here, click to see his full version), along with a graph for those whose minds work that way.
Lean practitioners and scientists have known for many decades that cost effectiveness isn't all it is cranked up to be. Cost effectiveness is a measure of how much of the capacity of a resource is utilized. The resource may be a person, a computer, an aircraft, a machine in a workshop...
One thing to watch out for in this discussion is the terminology. The terms effectiveness and efficiency are tossed about frequently, and it would be helpful to clarify. Effectiveness is a measure of whether the correct things are being done. Efficiency is a measure of how well or quickly work is being done. It doesn't necessarily measure whether the work is the most beneficial to the organization. However, it is usually much easier to measure efficiency: you just watch the machine or the person or the airplane and see if it is "busy." Henrik's "cost effectiveness" is an efficiency measure: how much "work" are you getting per unit.
The problem with efficiency measures is that they drive exactly the kind of behavior that Henrik describes in his post. If everyone must be at high utilization, then there is bound to be a breakdown due to Queuing Theory.
I suggest asking first whether the work being done is actually providing benefit to the organization. Is the work just there to boost efficiency numbers, or does it provide an overall benefit to the bottom line?